This invention relates to reaction hydraulic turbines. More specifically, this invention relates to reaction hydraulic turbines comprising a radial intake consisting of a spiral casing with stay vanes, a radial guide gate apparatus, and either a mixed-flow runner or an axial-flow runner.
In the prior art the reaction hydraulic turbines with radial intake and with either mixed-flow runner (Francis turbines) or axial-flow runner (Kaplan and propeller turbines) are commonly provided with a radial guide gate apparatus. The radial guide gate apparatus comprises a plurality of pivotal radial wicket gates whose pivots are parallel to the turbine axis. The radial guide gate apparatus regulates the flow of the water to the turbine runner and closes water passages when the turbine must be stopped. The runner blades must have a geometric shape that corresponds to the flow conditions at the runner inlet and exit in order to provide the requisite power at maximum efficiency. An example of such a Kaplan turbine with a radial intake is shown in U.S. Pat. No. 4,203,703.
Prior art wicket gates are cylindrical, i.e., are circular cylinders. The term "cylinder" is used herein and in the following disclosure in a strict mathematical sense to mean a three-dimensional geometric shape defined by a plurality of parallel lines. A cylinder having a constant radius cross section (which is commonly referred to simply as cylindrical) will, in this application, be referred to as a circular cylinder. From this strict definition of cylindrical, one can readily see that wicket gates of the prior art (such as those shown in U.S. Pat. No. 4,203,703) are cylinders in that they have a constant cross section (by a plane perpendicular to the turbine axis) and the gates' surfaces are defined by a plurality of parallel lines which are parallel to the rotary axis of the turbine runner. It is evident that the discharge angle at the trailing edge of these wicket gates from the bottom (or first end as used herein) of the gate to the top (or second end as used herein) is constant. "Discharge angle" means the angle between the trailing edge exit element and the radial direction. It will be appreciated that referring to the bottom (or first end as used herein) and top (or second end as used herein) of the gate implies a turbine having a vertical rotary axis. This is illustrative only and this disclosure is equally applicable to turbines having an inclined or horizontal axis.
Although some prior art radial gates vary from cylindrical at the top (in order to provide stress relief at the attachment to the stems), and some have thinner trailing edge at the bottom, for all prior art radial wicket gates the discharge angle at the trailing edge of the wicket gate from the top of the gate to the bottom is constant and its inlet and trailing edges are straight segments parallel to the turbine axis.
In bulb turbines, whose intake is not radial, the wicket gates do not necessarily have a constant discharge angle along the trailing edge. For example, in U.S. Pat. Nos. 4,120,602 and 3,973,869, the wicket gates extend radially away from the axis of the turbine and their pivots are not parallel to the turbine axis. To effect closure of the gate, these wicket gates are designed such that their leading and trailing edges are part of the same cone.
The constant discharge angle of prior art wicket gates results in a variable value of whirl, (V.sub.u R).sub.i-1 (V.sub.u is the circumferential component of absolute velocity vector R, is the radius from turbine axis) in the flow coming in the runner. This is due to the fact that water flows from the spiral casing through the wicket gates, the runner and the draft tube to discharge in a curved path.
Indeed, in turbines with a radial intake, the water flows radially through the wicket gates toward the turbine axis and is defect downwardly or axially toward the turbine discharge. Due to the curvature of the water flow, the radial component of the absolute velocity, V.sub.r, at the trailing edge of the wicket gate decreases from the bottom of the gate to the top of the gate. Where the curvature of the water path is very sharp, there can be a sharp decrease in V.sub.r from the bottom of the wicket gate to the top of the wicket gate.
The vector velocity of the water flow from the wicket gates follows the trailing edge exit element and has the same angle with radial direction from the top of the wicket gate to the bottom of the wicket gate. Since V.sub.r decreases from the bottom of the gate to the top of the gate, V.sub.u also gets smaller from the bottom of the wicket gate to the top of the wicket gate. Therefore, the value of whirl, (V.sub.u R).sub.i, decreases from the bottom to the top of the wicket gate (radius, R, along the discharge of the cylindrical wicket gate is constant).
The variable whirl value, (V.sub.u R).sub.i, in the flow coming to the runner causes the following phenomena at the design mode (at optimum):
1. The incoming flow is not potential. According to Stokes' theorem, variation of the whirl value (V.sub.u R).sub.i along the wicket gate trailing edge creates a vortex wake leaving the trailing edges along the streamlines of the absolute flow. Therefore, there are n.sub.wg vortex wakes in the flow coming to the runner (n.sub.wg is the number of the wicket gates).
2. The incoming flow is not axisymmetric, since the axisymmetry is destroyed by the n.sub.wg vortex wakes.
3. The flow leaving the runner at design mode has a positive whirl value, (V.sub.u R).sub.e, along the entire blade trailing edge, excluding the point at the hub/crown where the whirl value is zero; (V.sub.u R).sub.e has a maximum value at the periphery).
The change of the whirl value .DELTA.(V.sub.u R) in the runner for each elementary turbine is defined by the Euler equation: ##EQU1## where: g is the gravity acceleration
.eta..sub.e1 is the efficiency of the elementary turbine; PA1 H is the head of the turbine; and PA1 .omega.=.pi.N/30 is the angular velocity of turbine (N is the turbine rotation) PA1 V.sub.me is the value of meridional projection of absolute velocity along the runner blade trailing edge; PA1 L.sub.e is the total length of the blade trailing edge; PA1 dl.sub.e is differential of the length of the blade trailing edge; PA1 R.sub.e is the radius along the blade trailing edge; and PA1 Q is the flow rate of the turbine.
It is very important to note that for a turbine designed with .eta..sub.e1 =.eta..sub.max, .DELTA.(V.sub.u R) is constant along the runner span from the hub/crown to periphery.
The whirl value leaving the runner is: EQU (V.sub.u R).sub.e =(V.sub.u R).sub.i -.DELTA.(V.sub.u R) (2)
The value of (V.sub.u R).sub.e at the hub/crown must be equal to zero in order to avoid whirl along the turbine axis in the draft tube cone present after the runner, since the axial whirl causes loss of efficiency and instability of the flow in the draft tube. Thus, at the hub/crown the whirl value at the runner inlet is [(V.sub.u R).sub.i ].sub.h =.DELTA.(V.sub.u R). Since the whirl value at the runner inlet along the entire leading edge (excluding the point at the hub/crown) is larger than [(V.sub.u R).sub.i ].sub.h, the whirl value at the exit is positive and has a maximum at the periphery.
Let (V.sub.u R).sub.i =k[(V.sub.u R).sub.i ].sub.h, where k monotonically increases from k.sub.h =1, at the hub/crown, to k.sub.p =k.sub.m, at the periphery (according to the experimental data coefficient k.sub.m varies from 1.1 to 3.0 for different water passages geometries). Thus, the following conditions for the inlet and the exit of the runner exit: EQU (V.sub.u R).sub.i =k.DELTA.(V.sub.u R) (3) EQU (V.sub.u R).sub.e =(k.multidot.1).DELTA.(V.sub.u R) (4)
The phenomena described above are strongly pronounced in Kaplan, propeller and low/middle head Francis turbines (k.sub.m =2.0-3.0). In high head Francis turbines, these phenomena are not as strongly pronounced (k.sub.m =1.1-2.0), since in the high head Francis turbines the flow turns from radial to axial very gradually.
All these phenomena lead to the loss of efficiency. The energy of the vortices cannot be restored and they dissipate in the draft tube. The absence of the symmetry of the incoming flow causes instability of the flow angle along the runner blade inlet edge and, therefore, additional shock losses at the runner entrance are produced.
The losses causes by the positive value of the whirl along trailing edge of the runner can be computed as follows: ##EQU2## where: V.sub.ue is the value of circumferential projection of absolute velocity along the runner blade trailing edge;
The described losses are estimated to be up to 2% for Kaplan and propeller turbines and to be up to 4% for low/middle head Francis turbines.
Additionally, in the low/middle head Francis turbines, the high positive value of the whirl value at the blade exit at periphery causes the cavitation coefficient value .sigma. at periphery to be much higher than at the crown.